A equação de Schrödinger possui solução exata para átomos monoeletrônicos, porém devido ao nível de complexidade das soluções para átomos polieletrônicos e moléculas houve a necessidade de métodos aproximados que descrevessem satisfatoriamente as propriedades de interesse. O Método da Coordenada Geradora Hartree-Fock (MCG-HF) é um método variacional que permitiu desenvolver conjuntos de base acurados para átomos, entretanto, utilizando apenas a discretização integral, os conjuntos de base requeriam um maior número de expoentes, deixando-os extensos. Barbosa e da Silva (2009) introduziram o conceito de discretização integral polinomial que diminuiu a quantidade de expoentes necessários para descrever adequadamente um átomo, gerando conjuntos de base menores. Os conjuntos de primitivas gerados pelo p-MCG-HF foram contraídos utilizando o método de contração geral para a qualidade 5Z e em seguida, polarizados, utilizando dois métodos de polarização, ou seja, a pinçada e otimizada, para descrever melhor propriedades químicas como ligação química, estados excitados, entre outros. As energias obtidas por esses métodos de polarização são semelhantes, porém o tempo computacional é menor para a polarização pinçada (na maioria dos casos). Ao comparar os resultados obtidos neste trabalho com aqueles da literatura, os resultados obtidos utilizando o p-MCG-HF são de alta qualidade e com a enorme vantagem de ter tempo computacionais muito reduzidos. Esse método já está sendo aplicado por este grupo de pesquisa para gerar conjunto base para átomos do 1º ao 2º períodos da tabela periódica e os metais de transição e têm demonstrado resultados satisfatórios. Então, no intuito de estudar todos os átomos da tabela periódica até o bário, os átomos Gálio, Germânio, Arsênio, Selênio, Bromo, Criptônio, Rubídio, Estrôncio, Índio, Estanho, Antimônio, Telúrio, Iodo, Xenônio, Césio e Bário serão o foco deste trabalho. / The Schrödinger equation has exact solution for monoelectronic atoms, but the high complexity for the exact solution for many electron systems, lead to the use of approximation methods that satisfactorily. The Hartree-Fock Generating Coordinate Method (MCG-HF) is a variational method that allows the development of accurate basis sets for atoms with many electrons. However, using only an integral discretization, the basis sets generated with the MCG-HF requires a larger number of exponentes. Barbosa and da Silva (2009) introduced the concept of integral polynomial discretization that reduced the amount of exponents required to properly describe an atom, generating smaller basis sets. The sets of primitives generated with p-MCG-HF were contracted using the general contraction method for 5Z quality and then polarized using two different polarization methods (pinched and optimized) to better describe chemical properties such as chemical bonding, excited states, etc. The energies obtained with these polarization methods are similar, but the computational cost is smaller for pinched polarization (in most cases). Comparing the results obtained in this work with those of the literature, the results obtained by using the p-MCG-HF have a high quality and a large advantage of having very reduced computational cost. This method is already being applied in ours research group to generate basis set for atoms of the 1st to 2nd period of the periodic table and the transition metals showing satisfactory results. In order to study all the atoms in the periodic table, the atoms Gallium, Germanium, Arsenic, Selenium, Bromine, Crypton, Rubidium, Strontium, Indium, Tin, Antimony, Tellurium, Iodine, Xenium, Cesium and Barium will be focused in this work.

A teoria da perturbação baseada em múltiplas escalas temporais é usada neste trabalho para estudar as interações não lineares entre ondas na atmosfera no contexto das equações primitivas no plano beta-equatorial em coordenadas isobáricas. As ondas equatoriais de Rossby, mista Rossby-gravidade, gravidade-inerciais e Kelvin, com diferentes estruturas verticais, são obtidas como soluções de ordem dominante. A partir da condição de solvabilidade do problema de ordem superior, foi obtido um modelo assintótico reduzido que governa a interação fracamente não linear entre as ondas numa determinada tríade ressonante. A conservação da energia total para a solução de ordem dominante do modelo de equações original implica que a soma dos coeficientes de interação numa tríade ressonante qualquer deve ser nula. Usando um método gráfico, foram encontradas algumas tríades ressonantes envolvendo ondas baroclínicas equatorialmente confinadas e ondas de Rossby barotrópicas possuindo significativa influência em médias e altas latitudes. Duas tríades ressonantes receberam especial atenção devido ao significativo acoplamento entre os modos e à possível relação desses modos com alguns aspectos observados acerca dos fenômenos de baixa freqüência na atmosfera. A primeira tríade é composta por uma onda de Rossby barotrópica possuindo o segundo modo meridional interagindo ressonantemente com uma onda mista de Rossby-gravidade com a estrutura vertical do primeiro modo baroclínico, ambas possuindo o mesmo número de onda zonal, por meio de um modo geostrófico zonalmente simétrico (k = 0) possuindo a mesma estrutura vertical da onda mista e o modo meridional n = 1. A segunda tríade é constituída por uma onda mista de Rossby-gravidade com número de onda-2, uma onda de Kelvin com número de onda-1, ambas associadas ao primeiro modo baroclínico, e uma onda de Rossby barotrópica com número de onda-3 e modo meridional n = 2. A integração das equações reduzidas para essas duas tríades ressonantes mostra que para as amplitudes iniciais dos modos caracterizando magnitudes típicas de anomalias atmosféricas observadas, os modos na primeira tríade, em geral, trocam energia na escala intra-sazonal, enquanto os modos na segunda tríade trocam energia na escala intra-sazonal ou numa escala semi-anual, dependendo da amplitude inicial do modo de Rossby barotrópico. São discutidas as implicações dos resultados para a dinâmica da interação trópicos-extratrópicos na escala de baixa-freqüência do espectro dos movimentos atmosféricos. / The asymptotic perturbation theory based on multiple scales in time is used to investigate the nonlinear interactions among equatorial waves in an equatorial beta-plane adiabatic primitive equation atmospheric model in isobaric coordinates. The equatorial Rossby, mixed Rossby-gravity, Kelvin and inertio-gravity waves, with several vertical structures, are obtained as the leading-order solution. From the solvability condition of the problem, a reduced model governing the weakly nonlinear interaction of the waves in a particular resonant triad was obtained. The total energy conservation of the leading-order solution of the original model equations implies that the sum of the coupling coefficients in any resonant triad must be zero. Using a graphical approach, we determined some resonant triads involving equatorially trapped baroclinic waves and barotropic Rossby waves having large mid-latitude amplitude. Two particular interactions deserve special attention because of their strong coupling and their possible relation to observed features in the atmospheric circulation. One is characterized by a first baroclinic mode structure mixed Rossby-gravity wave interacting with a barotropic Rossby wave with the second gravest meridional mode, both having the same wavenumber, through a zonally symmetric geostrophic mode with the same vertical structure as the Yanai wave and having the n = 1 meridional mode. The other triad is composed of a zonal wavenumber-1 Kelvin wave, a zonal wavenumber-2 mixed Rossby-gravity wave, both with the first baroclinic mode vertical structure, and a barotropic zonal wavenumber-3 Rossby wave having the second gravest meridional mode. The barotropic Rossby waves in these two triad interactions have significant projection onto middle and higher latitudes. The integration of the triad equations for these particular interactions shows that, for the initial mode amplitudes characterizing typical magnitudes of atmospheric flow perturbations, the modes in the first triad usually exchange energy on intraseasonal time-scales, while the modes in the second triad exchange energy on either intraseasonal or semi-annual time-scale, depending on the initial condition. The implications of the results for the dynamics of tropics-extratropics interaction on low-frequency time-scales are discussed.

Nesse trabalho, abordamos as condições de reprodução do campesinato no Vale do Jequitinhonha mineiro, da sua formação até os dias atuais, tomando-as como momento da territorialização do capital e da mobilização do trabalho, observadas sempre nas transformações que sofreram no curso contraditório do processo de modernização. Investigamos a formação regional do campesinato no bojo da transição do escravismo colonial para o trabalho livre, relação engendrada como desdobramento da mineração ocorrida em muitos afluentes da bacia do rio Jequitinhonha e do estabelecimento e da expansão das fazendas pecuárias no que outrora fora considerado \"sertão\". Analisamos também a forma de reprodução desse campesinato, tomando-a como uma relação social de produção na qual se assentou a reprodução do capital, quando a mesma ainda não podia prescindir do domínio fundiário e recurso ao exercício da violência por parte daqueles que personificavam o capital e da produção direta dos meios de vida por parte daqueles que personificavam o trabalho. Apresentamos ainda a acumulação das condições para o rompimento daquela relação social de produção como resultado central da própria territorialização do capital responsável por engendrá-la, entre elas, inclusive, a institucionalização do Estado e a formação da sua tecnocracia, ocorridas, ambas, em meio ao processo de autonomização das categorias terra, trabalho e capital, o qual investigamos a partir da intervenção do planejamento regional estatal no Vale do Jequitinhonha e das invasões e expulsões de agregados, posseiros e situantes que foram desencadeadas especialmente nas décadas de 1960, 1970 e 1980. Na análise dos desdobramentos dessa ruptura enfatizamos a permanência do campesinato no Vale do Jequitinhonha, porém, em meio a uma profunda transformação nas relações sociais de produção em que o mesmo se encontrava engajado, as quais passaram a se assentar na generalização da mobilidade do trabalho. Por fim, as condições atuais de reprodução do campesinato, interpretamos como momento da reprodução do capital em sua crise fundamental. Articulamos a exposição dos processos apontados com uma discussão sobre o papel da acumulação primitiva na reprodução do capital e as limitações para sua reiteração continuada; sobre a homogeneização e a diferenciação das relações sociais de produção; sobre o caráter contraditório e fundamentalmente crítico do desenvolvimento capitalista e o caráter da sua crise atual. Conduzimos a mesma a partir da problematização de inúmeros estudos dedicados ao problema da reprodução camponesa no Vale do Jequitinhonha, criticando a apreensão que faziam dessa última como totalidade apartada por não reconhecerem ser essa aparência resultante do processo de autonomização. Analisamos ainda um farto conjunto de depoimentos de lavradores e lavradoras no qual os mesmos discutiam transformações experimentadas em suas condições de reprodução, buscando articular, igualmente, os planos da história e da experiência, a partir de uma crítica do processo de sujeição dos sujeitos sociais a uma dominação abstrata, fetichista e tautológica da mercadoria enquanto forma de mediação e do capital enquanto sujeito automático. / On this thesis we approached the reproduction conditions of peasants from Jequitinhonha Valley, in Minas Gerais, Brazil, from its formation until nowadays. We grasped those reproduction conditions as territorialization of capital and labor mobilization moments, moments that we observed always on the transformations that had occurred on the contradictory process of modernization. We researched regional peasantry formation from the enslavement transition to free labor, a relation that was engendered as the unfold of mining in affluent rivers of Jequitinhonha river and of cattle farming establishment and expansion in what was once considered sertão. We also analyzed the reproduction form of this peasantry as a social relation of production which was the basis of capital reproduction in a moment that personified capital could not prescind from land domination and from violence exercise. After that, we present the accumulation of conditions that ruptured that social relation of production as a central result of territorialization of capital itself, conditions such as State institutionalization and the formation of its technocracy, both occurred throughout autonomization process of land, labor and capital categories. We researched that process from the intervention of regional State planning on the Jequitinhonha Valley and from invasions and expulsions of agregados, posseiros e situantes that occurred specially on the decades of 1960, 1970 and 1980. While analyzing the unfolding of that rupture we give emphasis on the permanence of peasants on the Jequitinhonha Valley, although, in the middle of a deep transformation of social relations of production that these peasants were engaged, which passed to be embedded on the generalization of labor mobilization. Finally, we interpreted actual peasantry reproduction conditions as the reproduction of capital in its fundamental crisis. We articulated the exposition of the processes already mentioned with a discussion of the role of primitive accumulation for capital accumulation and the limits to its continuous reiteration; on the homogenization and differentiation of social relations of production; and on the contradictory and fundamentally critical character of capitalistic development and its actual crisis character. We conducted such issue questioning numerous researches that were dedicated to the peasantry reproduction problem on the Jequitinhonha Valley, and we criticized the grasping of that social reproduction as a separated totality as those researches didnt recognized that as an appearance of autonomization process. We also analyzed a big amount of testimonials of lavradores and lavradoras, in which they discussed transformations experienced on their conditions of reproduction, as we tried to articulate historical and experience plans, from a critique of the process of social subjects subjection by an abstract, fetishistic and tautological domination of merchandise form as an automatic subject mediation and capital form.

Nous proposons dans cette thèse une nouvelle approche cumulative de recalage d'images basée sur des primitives construites à partir des lignes de niveaux. Les lignes de niveaux sont invariantes par rapport aux diverses perturbations affectant l'image tels que les changements de contraste. Par ailleurs, leur abondance dans une image suggère naturellement un processus de décision cumulatif. Nous proposons alors un algorithme récursif d'extraction des lignes de niveaux simple et efficace qui extrait les lignes par groupes rectiligne appelés ``segments''. Les segments sont ensuite groupés -- sous contrainte de proximité -- en fonction du modèle de transformation recherchée et afin de faciliter le calcul des invariants. Les primitives construites ont alors la forme de Z, Y ou W et sont classées en fonction de leur fiabilité, ce qui participe au paramétrage du processus de décision cumulatif. Le vote est multi-tours et constitué d'une phase préliminaire de construction de listes de préférences inspiré de la technique des mariages stables. Les primitives votent à une itération donnée en fonction de leur fiabilité. Chaque itération fournit ainsi un estimé de la transformation recherchée que le tour suivant peut raffiner. Ce procédé multi-tours permet, de ce fait, d'éliminer les ambiguïtés d'appariement générées par les motifs répétitifs présents dans les images. Notre approche a été validée pour recaler des images sous différents modèles de transformations allant de la plus simple (similarité) à la plus complexe (projective). Nous montrons dans cette thèse comment le choix pertinent de primitives basées sur les lignes de niveaux en conjonction avec un processus de décision cumulatif permet d'obtenir une méthode de recalage d'images robuste, générique et complète, fournissant alors différents niveaux de précision et pouvant ainsi s'appliquer à différents contextes. / In this thesis, we propose a new image registration method that relies on level-line primitives. Level-lines are robust towards contrast changes and proposed primitives inherit their robustness. Moreover, their abundance in the image is well adapted to a cumulative matching process based on a multi-stage primitive election procedure. We propose a simple recursive tracking algorithm to extract level lines by straight sets called "segments". Segments are then grouped under proximity constraints to construct primitives (Z, Y and W shapes) that are classified into categories according to their reliability. Primitive shapes are defined according to the transformation model. The cumulative process is based on a preliminary step of preference lists construction that is inspired from the stable marriage matching algorithm. Primitives vote in a given voting stage according to their reliability. Each stage provides a coarse estimate of the transformation that the next stage gets to refine. This process, in turn, eliminate gradually the ambiguity happened by incorrect correspondences. Our additional contribution is to validate further geometric transformations, from simple to complex ones, completing the path "similarity, affine, projective". We show in this thesis how the choice of level lines in conjunction with a cumulative decision process allows defining a complete robust registration approach that is tested and evaluated on several real image sequences including different type of transformations.

Medulloblastoma (MB) and supratentorial primitive neuroectodermal tumor (stPNET) are pediatric embryonic brain tumors, which arise in a brain that is in the process of growth and development. They differ significantly from adult lesions and may involve unique genetic and epigenetic factors. However, the pathogenesis of these tumors is still elusive. My project consisted of four parts, investigating major genetic and epigenetic alterations of these tumors. / Multiple genetic studies have shown high frequency of loss (30--60%) on chromosome 8p in MBs. Microcell-mediated transfer of chromosome 8 suppressed tumorigenesis or the proliferation of colon and breast cancer cell, indicating that chromosome 8p is likely to include several TSGs in human cancers. In previous studies from our laboratory, results showed the frequency of loss on chromosome 8p is also rather high (66.7%). An overlapping HD region was identified in a 1.8cM interval on 8p22-23.1, between markers D8S520 and D8S1130, in two MBs (Yin et al., 2002), indicating that several candidate TSGs are located within or near this region. PinX1 on 8p23.1, a potential inhibitor of telomerase, is most likely the candidate TSG in MBs due to its location and function. To evaluate the genetic alterations of PinX1 and to investigate its role in MBs, the first part of my study is to perform mutation analysis in a series of 52 primary MBs, 3 MB cell lines and 4 primary stPNETs. Transcript expression of PinX1 was evaluated by reverse transcription-polymerase chain reaction (RT-PCR) in microdissected tumors and normal cerebellum. Using the telomeric repeat amplification protocol (TRAP) assay, 19 MBs, 2 stPNETs and all 3 MB cell lines were analyzed for telomerase activity. No somatic point mutations and loss of expression of PinX1 were detected in our series, suggesting that PinX1 is not the target gene on 8p23.1 in MBs. Although we did not find a significant association between PinX1 expression and telomerase activity, the presence of telomerase activity in 16 of 22 MBs and 1 of 2 stPNETs indicate that telomerase activation is associated with the development of this malignant disease. Our study represents the largest series of MB examined by telomerase repeat amplification protocol (TRAP) assay. (Abstract shortened by UMI.) / Chang Qing. / "April 2005." / Adviser: Ho-Keung Ng. / Source: Dissertation Abstracts International, Volume: 67-01, Section: B, page: 0191. / Thesis (Ph.D.)--Chinese University of Hong Kong, 2005. / Includes bibliographical references (p. 201-228). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Electronic reproduction. [Ann Arbor, MI] : ProQuest Information and Learning, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts in English and Chinese. / School code: 1307.

No description available.

Au cours de la préimplantation, l'embryon de souris produit deux lignages cellulaires, le trophectoderme (TE), et la masse cellulaire interne (MCI) qui elle-même se différencie en épiblaste (Epi) et en endoderme primitif (EPr), caractérisés respectivement par l'expression exclusive de Nanog et de Gata6. La voie FGF/MAPK joue un rôle critique dans l’acquisition de l’identité EPr. J’ai examiné l’expression de pERK, DUSP4 et ETV5 qui permettent de visualiser l'activité des MAPK. Ces analyses ont été effectuées en activant ou inhibant la voie FGF/MAPK, ainsi que dans des embryons mutants pour Nanog et/ou Gata6. Ceci a permis d’observer l’activation de la voie FGF/MAPK dès E3,25. Un autre volet de mon travail a été d'analyser la voie de l’IGF dans les embryons préimplantatoires afin de comprendre l’influence de cette voie dans les différents lignages. J’ai montré que le récepteur activé pIGF1R est exprimé de manière différentielle dans le TE, l’EPr et l’Epi au cours du développement. Une supplémentation d’IGF1 induit une augmentation du nombre de cellules en deux phases, d'abord de l’Epi puis de l’EPr. A l’inverse, une perte de fonction d’IGF1R induit une diminution du nombre de cellules entre E3,75 et E4,25. / During preimplantation, mouse embryo produces two cellular lineages, the trophectoderm (TE), and the inner cell mass (ICM), which differentiates in epiblast (Epi) and primitive endoderm (PrE), characterized respectively by the complementary expression of Nanog and Gata6. FGF/MAPK pathway plays a critical role in the acquisition of a PrE identity. I examined the expression of the markers of MAPK activity pERK, DUSP4 and ETV5. The analyze was performed with activation or inhibition of FGF/MAPK pathway and in mutant embryos for Nanog or Gata6. This showed that FGF/MAPK pathway is activated as soon as E3,25. I have also analyzed the IGF pathway in preimplantation embryos in order to understand the role of this pathway in embryonic lineages. I showed that active receptor pIGF1R is differentially expressed in TE, PrE and Epi during embryonic development. Supplementation with IGF1 induces an increase in cell number in two phases, first in Epi then in PrE. Conversely, loss of function of IGF1R induces a decrease in cell number between E3,75 and E4,25.

The ability to learn is essential for robots to function and perform services within a dynamic human environment. Robot programming by demonstration facilitates learning through a human teacher without the need to develop new code for each task that the robot performs. In order for learning to be generalizable, the robot needs to be able to grasp the underlying structure of the task being learned. This requires appropriate knowledge abstraction and representation. The goal of this thesis is to develop a learning by imitation system that abstracts knowledge of human demonstrations of a task and represents the abstracted knowledge in a hierarchical framework. The learning by imitation system is capable of performing both action and object recognition based on video stream data at the lower level of the hierarchy, while the sequence of actions and object states observed is reconstructed at the higher level of the hierarchy in order to form a coherent representation of the task. Furthermore, error recovery capabilities are included in the learning by imitation system to improve robustness to unexpected situations during task execution. The first part of the thesis focuses on motion learning to allow the robot to both recognize the actions for task representation at the higher level of the hierarchy and to perform the actions to imitate the task. In order to efficiently learn actions, the actions are segmented into meaningful atomic units called motion primitives. These motion primitives are then modeled using dynamic movement primitives (DMPs), a dynamical system model that can robustly generate motion trajectories to arbitrary goal positions while maintaining the overall shape of the demonstrated motion trajectory. The DMPs also contain weight parameters that are reflective of the shape of the motion trajectory. These weight parameters are clustered using affinity propagation (AP), an efficient exemplar clustering algorithm, in order to determine groups of similar motion primitives and thus, performing motion recognition. The approach of DMPs combined with APs was experimentally verified on two separate motion data sets for its ability to recognize and generate motion primitives. The second part of the thesis outlines how the task representation is created and used for imitating observed tasks. This includes object and object state recognition using simple computer vision techniques as well as the automatic construction of a Petri net (PN) model to describe an observed task. Tasks are composed of a sequence of actions that have specific pre-conditions, i.e. object states required before the action can be performed, and post-conditions, i.e. object states that result from the action. The PNs inherently encode pre-conditions and post-conditions of a particular event, i.e. action, and can model tasks as a coherent sequence of actions and object states. In addition, PNs are very flexible in modeling a variety of tasks including tasks that involve both sequential and parallel components. The automatic PN creation process has been tested on both a sequential two block stacking task and a three block stacking task involving both sequential and parallel components. The PN provides a meaningful representation of the observed tasks that can be used by a robot to imitate the tasks. Lastly, error recovery capabilities are added to the learning by imitation system in order to allow the robot to readjust the sequence of actions needed during task execution. The error recovery component is able to deal with two types of errors: unexpected, but known situations and unexpected, unknown situations. In the case of unexpected, but known situations, the learning system is able to search through the PN to identify the known situation and the actions needed to complete the task. This ability is useful not only for error recovery from known situations, but also for human robot collaboration, where the human unexpectedly helps to complete part of the task. In the case of situations that are both unexpected and unknown, the robot will prompt the human demonstrator to teach how to recover from the error to a known state. By observing the error recovery procedure and automatically extending the PN with the error recovery information, the situation encountered becomes part of the known situations and the robot is able to autonomously recover from the error in the future. This error recovery approach was tested successfully on errors encountered during the three block stacking task.

In 1878, Lucas \cite{lucas} investigated the sequences $(\ell_n)_{n=0}^\infty$ where $$\ell_n=\frac{\alpha^n-\beta^n}{\alpha-\beta},$$ $\alpha \beta$ and $\alpha+\beta$ are coprime integers, and where $\beta/\alpha$ is not a root of unity. Lucas sequences are divisibility sequences; if $m|n$, then $\ell_m|\ell_n$, and more generally, $\gcd(\ell_m,\ell_n)=\ell_{\gcd(m,n)}$ for all positive integers $m$ and $n$. Matijasevic utilised this divisibility property of Lucas sequences in order to resolve Hilbert's 10th problem. \noindent In 1930, Lehmer \cite{lehmer} introduced the sequences $(u_n)_{n=0}^\infty$ where \begin{eqnarray*} u_n& = & \frac{\alpha^{n}-\beta^n}{\alpha^{\epsilon(n)}-\beta^{\epsilon(n)}},\\ \epsilon(n)&=&\left\{\begin{array}{ll} 1, \hspace{.1in}\mbox{if}\hspace{.1in}n\equiv 1 \pmod 2;\\ 2, \hspace{.1in}\mbox{if}\hspace{.1in}n\equiv 0\pmod 2;\end{array}\right. \end{eqnarray*} $\alpha \beta$ and $(\alpha +\beta)^2$ are coprime integers, and where $\beta/\alpha$ is not a root of unity. The sequences $(u_n)_{n=0}^\infty$ are known as Lehmer sequences, and the terms of these sequences are known as Lehmer numbers. Lehmer showed that his sequences had similar divisibility properties to those of Lucas sequences, and he used them to extend the Lucas test for primality. \noindent We define a prime divisor $p$ of $u_n$ to be a primitive divisor of $u_n$ if $p$ does not divide $$(\alpha^2-\beta^2)^2u_3\cdots u_{n-1}.$$ Note that in the list of prime factors of the first $n-1$ terms of the sequence $(u_n)_{n=0}^\infty$, a primitive divisor of $u_n$ is a new prime factor. \noindent We let \begin{eqnarray*} \kappa& = & k(\alpha \beta\max\{(\alpha-\beta)^2,(\alpha+\beta)^2\}),\\ \eta & = & \left\{\begin{array}{ll}1\hspace{.1in}\mbox{if}\hspace{.1in}\kappa\equiv 1\pmod 4,\\ 2\hspace{.1in}\mbox{otherwise},\end{array}\right. \end{eqnarray*} where $k(\alpha \beta \max\{(\alpha-\beta)^2,(\alpha+\beta)^2\})$ is the squarefree kernel of $\alpha \beta \max\{(\alpha-\beta)^2,(\alpha+\beta)^2\}$. On the one hand, building on the work of Schinzel \cite{schinzelI}, we prove that if $n>4$, $n\neq 6$, $n/(\eta \kappa)$ is an odd integer, and the triple $(n,\alpha,\beta)$, in case $(\alpha-\beta)^2>0$, is not equivalent to a triple $(n,\alpha,\beta)$ from an explicit table, then the $n$th Lehmer number $u_n$ has at least two primitive divisors. Moreover, we prove that if $n\geq 1.2\times 10^{10}$, and $n/(\eta \kappa)$ is an odd integer, then the $n$th Lehmer number $u_n$ has at least two primitive divisors. On the other hand, building on the work of Stewart \cite{stewart77}, we prove that there are only finitely many triples $(n,\alpha,\beta)$, where $n>6$, $n\neq 12$, and $n/(\eta \kappa)$ is an odd integer, such that the $n$th Lehmer number $u_n$ has less than two primitive divisors, and that these triples may be explicitly determined. We determine all of these triples $(n,\alpha,\beta)$ up to equivalence explicitly when $6<n\leq 30$, $n\neq 12$, and $n/(\eta \kappa)$ is an odd integer, and we tabulate the triples $(n,\alpha,\beta)$ we discovered, up to equivalence, for $30<n\leq 500$. Finally, we show that the conditions $n>6$, $n\neq 12$, are best possible, subject to the truth of two plausible conjectures.

In 1878, Lucas \cite{lucas} investigated the sequences $(\ell_n)_{n=0}^\infty$ where $$\ell_n=\frac{\alpha^n-\beta^n}{\alpha-\beta},$$ $\alpha \beta$ and $\alpha+\beta$ are coprime integers, and where $\beta/\alpha$ is not a root of unity. Lucas sequences are divisibility sequences; if $m|n$, then $\ell_m|\ell_n$, and more generally, $\gcd(\ell_m,\ell_n)=\ell_{\gcd(m,n)}$ for all positive integers $m$ and $n$. Matijasevic utilised this divisibility property of Lucas sequences in order to resolve Hilbert's 10th problem. \noindent In 1930, Lehmer \cite{lehmer} introduced the sequences $(u_n)_{n=0}^\infty$ where \begin{eqnarray*} u_n& = & \frac{\alpha^{n}-\beta^n}{\alpha^{\epsilon(n)}-\beta^{\epsilon(n)}},\\ \epsilon(n)&=&\left\{\begin{array}{ll} 1, \hspace{.1in}\mbox{if}\hspace{.1in}n\equiv 1 \pmod 2;\\ 2, \hspace{.1in}\mbox{if}\hspace{.1in}n\equiv 0\pmod 2;\end{array}\right. \end{eqnarray*} $\alpha \beta$ and $(\alpha +\beta)^2$ are coprime integers, and where $\beta/\alpha$ is not a root of unity. The sequences $(u_n)_{n=0}^\infty$ are known as Lehmer sequences, and the terms of these sequences are known as Lehmer numbers. Lehmer showed that his sequences had similar divisibility properties to those of Lucas sequences, and he used them to extend the Lucas test for primality. \noindent We define a prime divisor $p$ of $u_n$ to be a primitive divisor of $u_n$ if $p$ does not divide $$(\alpha^2-\beta^2)^2u_3\cdots u_{n-1}.$$ Note that in the list of prime factors of the first $n-1$ terms of the sequence $(u_n)_{n=0}^\infty$, a primitive divisor of $u_n$ is a new prime factor. \noindent We let \begin{eqnarray*} \kappa& = & k(\alpha \beta\max\{(\alpha-\beta)^2,(\alpha+\beta)^2\}),\\ \eta & = & \left\{\begin{array}{ll}1\hspace{.1in}\mbox{if}\hspace{.1in}\kappa\equiv 1\pmod 4,\\ 2\hspace{.1in}\mbox{otherwise},\end{array}\right. \end{eqnarray*} where $k(\alpha \beta \max\{(\alpha-\beta)^2,(\alpha+\beta)^2\})$ is the squarefree kernel of $\alpha \beta \max\{(\alpha-\beta)^2,(\alpha+\beta)^2\}$. On the one hand, building on the work of Schinzel \cite{schinzelI}, we prove that if $n>4$, $n\neq 6$, $n/(\eta \kappa)$ is an odd integer, and the triple $(n,\alpha,\beta)$, in case $(\alpha-\beta)^2>0$, is not equivalent to a triple $(n,\alpha,\beta)$ from an explicit table, then the $n$th Lehmer number $u_n$ has at least two primitive divisors. Moreover, we prove that if $n\geq 1.2\times 10^{10}$, and $n/(\eta \kappa)$ is an odd integer, then the $n$th Lehmer number $u_n$ has at least two primitive divisors. On the other hand, building on the work of Stewart \cite{stewart77}, we prove that there are only finitely many triples $(n,\alpha,\beta)$, where $n>6$, $n\neq 12$, and $n/(\eta \kappa)$ is an odd integer, such that the $n$th Lehmer number $u_n$ has less than two primitive divisors, and that these triples may be explicitly determined. We determine all of these triples $(n,\alpha,\beta)$ up to equivalence explicitly when $6<n\leq 30$, $n\neq 12$, and $n/(\eta \kappa)$ is an odd integer, and we tabulate the triples $(n,\alpha,\beta)$ we discovered, up to equivalence, for $30<n\leq 500$. Finally, we show that the conditions $n>6$, $n\neq 12$, are best possible, subject to the truth of two plausible conjectures.